The above is the clear rejection of the ultimate reality of ‘self’ or ‘soul’ by the Buddha, based on the “five aggregates”: no soul can be found that would be limited to one of the aggregates, would ‘own’ any of the aggregates, would include, or be included by, any of the aggregates.

Your body might change without your “sense of self” being affected; your feelings might change without your “sense of self” being affected; what you perceive… your convictions and habits, what you think… and what you experience might change without your “sense of self” being affected.
But as importantly, your “sense of self” cannot just decide what body to have (e.g. not to age… or without the pain…), your “sense of self” cannot just decide what feelings to have, what perception to have, what ideas to have, what experience t have. The ‘self’ doesn't own or control the aggregates.

People might make assumptions about the soul, but it cannot be found by a valid mean of cognition, so Buddhism considers the ‘self’ exists only as a convention, as a concept… which might prove useful at times but does not exist as an ultimate basis of reality, a building block we can ‘rely’ on (thanks to clear characteristics and traits) when analysing other phenomena.

So far, so standard…

Wholesome views, skilful views

Please note though that it was not said “the self doesn't exist”.
It was said “the self exists only as a convention”. The nuance is important because views exist, they inform some of our actions (i.e. they have causal efficacy), they inform some of our anticipations and trigger some of our disappointments (when anticipations don't prove reliable)… Views are major causes of unsatisfactoriness, and views about the self particularly so. To believe that “no self exists (at all)” is rejected by Buddhism as strongly as to believe that “a self ultimately exists, separate from the five aggregates.”

Views on the self and on the aggregates easily bind people into suffering:

One of the causes of suffering is that, if you erroneously believe your ‘self’ to be an aggregate, be “in” an aggregate, include an aggregate, or be separate but own an aggregate… then, whatever limitations you might (correctly or erroneously) associate to the aggregate, you will erroneously generalise to “who you are”.

In fact, the aggregates do not limit your buddha-nature so, whatever limitations you might associate to the said aggregates, these limitations do not constitute ‘you’! You can be a buddha!
As an aside, any sexist view claiming that women are inferior —because of the body-aggregate they currently identify with— is mistaken: it falls precisely for what the Buddha rejected! No woman is spiritually limited by her body! While she might limit herself by clinging to her body, she's no more limited by this than men are by their respective clinging.

Views on the lack of inherent self easily bind people into suffering too, because it easily feeds e.g. pessimism and nothing is done to improve the situation at hand so of course it doesn't improve… It is crucial to avoid extremes.
Just like the soul, buddha-nature cannot be isolated, pinpointed, nor found… which doesn't mean that it doesn't exist at all. If you fall for nihilism about the soul, you will also fall for nihilism about buddha-nature and about nirvāṇa itself. Nihilism is not a skilful view.
The “soul” is a concept, “buddha” is a concept too: the latter is just more useful and skilful than the former. “buddha” is so much a concept that a Zen master told “if you meet the buddha, kill him!" Stop imagining it, be it! Without a self, you're not missing anything; on the contrary, thus, no self ties you down to not being a buddha.

An important difficulty about the anatta (“no intrinsic self”) doctrine is how people easily fall back into nihilism or essentialism, directly or indirectly.
The direct nihilism, “I don't have a(ny) self”, is easy to see: just smack the head of the person and when (s)he complains, ask “who's complaining?"
The indirect essentialism is harder to see, and many buddhists think they have figured out a ‘truth’ when they cling to such a view: it lies in the “I am the experiencer, I cannot find the body which'd be ‘mine’ ” view (a variant might be “I don't find any essential gender as ‘mine’ ” when working on the ‘emptiness’ of gender). If you see yourself solely as the experiencer that you somehow conceptualise separate from the aggregates, you've fallen into a new version of the ‘soul’ concept: there is no experiencer separate from body, feeling, perception, views and consciousness!

You are no other than the five aggregates. Your self is not your body, not your feelings, not your perceptions, not your views, nor your consciousness… it doesn't exist separately from these, it doesn't control these (e.g. you cannot just chose to change body or to stop ageing), it isn't controlled by these… but it also is no other than these five aggregates.

So you're not your body, but you're also not not-your-body… and you're neither body-and-not-body, nor neither-body-nor-not-body!

The key question is: how do you relate to these aggregates?
Do you cling to them?
Do you project permanencies on them?
Do you get upset when ‘external’ factors (“not mine”) influence them?

Don't deny yourself the path of cultivation!

The path of cultivation has to be embodied, not just observed. Work on your views, act and speak wholesomely, guard your sense-doors and be mindful… If you're just an observer, how will you do that?

You're not your body, you're not not-your-body… and you're neither body-and-not-body, nor neither-body-nor-not-body!

You cannot control your body enough to pretend that it's “yours”, that a self ‘owns’ it… but you influence it enough to embody wisdom.
You cannot control your perception enough to pretend that a self ‘owns’ it and can unilaterally decide what's perceived and what's not… but you influence it enough (via its dependency on views, and ‘attitude’) to wisely discern rather than ignorantly discern.

Don't deny yourself the path of cultivation: no soul can be found that would be limited to one of the aggregates, would ‘own’ any of the aggregates, would include, or be included by, any of the aggregates.
But inter-dependency and co-dependent arising mean you cannot assert that you're not your aggregates.
So, drop the search for “who you are” (trying to find a satisfactory representation you can cling to), and be who you are! And since you have a choice about it, wisely be your aggregates, as wisely as you can! And learn through all of your aggregates, to see reality as it is.

Don't just stay in your head (merely reducing Buddhism to replacing old views by Buddhist views)! Generosity, for example, means little if it's limited to understanding and intention only. Forgiveness… Patience… Perseverance… Non-reactivity… Wisdom…
If understanding causality and Wisdom doesn't influence your body, your feelings, your perception, your views, your consciousness, what is Wisdom? And what is the point? Don't deny yourself the path of cultivation: you're not with a self, but you're not without a self either (and you're not with-and-without a self, you're not neither-with-nor-without a self). Drop the search for ‘your’ self (trying to find a satisfactory representation you can cling to), and be a buddha! Be a self-less True Self!

Moment after moment, what's the appropriate response (without biases, preconceptions, prejudices, preferences) to the situation at hand? What's the appropriate bodily response? What's the way to perceive things constructively, how many perspectives are relevant, what's the appropriate big picture and the appropriate attention to details? What's the appropriate emotional response? What's the appropriate intention?